1. 1
   The quadratic function $f$ is defined by $f(x) = 2x^2 - 20x + 44$, for $x \in \mathbb{R}$.

   (a)

   Write down the value of $f(0)$.

   [1 mark]

   • A$44$
   • B$20$
   • C$-20$
   • D$2$

   (b)

   Express $f(x)$ in the form $a(x-h)^2+k$. Find the value of $k$.

   [1 mark]

   • A$44$
   • B$-6$
   • C$-50$
   • D$6$

   (c)

   Hence find the exact solutions of $f(x)=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x) = \dfrac{4x-9}{x-3}$, for $x\in\mathbb{R}$, $x\neq3$.

   (a)

   Write down the equation of the vertical asymptote of the graph of $y=g(x)$.

   [1 mark]

   • A$x=4$
   • B$y=3$
   • C$x=3$
   • D$x=-3$

   (b)

   Write down the equation of the horizontal asymptote of the graph of $y=g(x)$.

   [1 mark]

   • A$y=-9$
   • B$y=-3$
   • C$x=4$
   • D$y=4$

   (c)

   Hence find the coordinates of the points where the graph of $y=g(x)$ crosses the coordinate axes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the quadratic equation $3kx^2 + 2x + k = 0$, where $k\in\mathbb{R}$ and $k\neq0$.

   (a)

   Find the set of values of $k$ for which the equation has two distinct real roots.

   [3 marks]

   (b)

   For the value $k=\dfrac13$, solve the equation $3kx^2+2x+k=0$ exactly.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x) = x^2 - 4x + 7$, for $x\in\mathbb{R}$.

   (a)

   (i) Express $f(x)$ in the form $(x-p)^2+q$. [2]
   (ii) The graph of $y=f(x)$ is transformed to the graph of $y=g(x)$ by a translation of $\begin{pmatrix}3\\-5\end{pmatrix}$ followed by a vertical stretch with scale factor 2 relative to the $x$-axis. Find $g(x)$ in the form $a(x-h)^2+k$. [4]

   [6 marks]

   (b)

   The function $h$ is defined by $h(x) = 3f(x) - 1$, restricted to the domain $x \ge 2$.
   (i) Explain why $h$ has an inverse function on this domain. [1]
   (ii) Find $h^{-1}(x)$ and state its domain. [5]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $f$ is defined by $f(x) = 3^{x} + 1$, for $x\in\mathbb{R}$.

   (a)

   Write down the equation of the horizontal asymptote of the graph of $y=f(x)$.

   [1 mark]

   • A$y=1$
   • B$y=0$
   • C$y=3$
   • D$x=0$

   (b)

   Find the value of $x$ for which $f(x)=28$.

   [1 mark]

   • A$9$
   • B$3$
   • C$\log_3 28$
   • D$2$

   (c)

   Hence find $f^{-1}(x)$ and state its domain.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $h$ is defined by $h(x) = x^3 - 7x + 6$, for $x\in\mathbb{R}$.

   (a)

   Which of the following is a root of $h(x)=x^3-7x+6$?

   [1 mark]

   • A$3$
   • B$-1$
   • C$1$
   • D$0$

   (b)

   Hence write down the sum of the other two roots of $h(x)=0$.

   [1 mark]

   • A$1$
   • B$7$
   • C$-7$
   • D$-1$

   (c)

   Hence find the other two roots of $h(x)=0$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The function $r$ is defined by $r(x) = \dfrac{2x^2-5x-3}{x-4}$, for $x\in\mathbb{R}$, $x\neq4$.

   (a)

   Show that $r(x) = 2x+3+\dfrac{9}{x-4}$, and hence write down the equation of the oblique asymptote of the graph of $y=r(x)$.

   [3 marks]

   (b)

   Hence, using the result from part (a), solve the inequality $r(x) \ge 2x+3$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A factory manufactures metal rods with a target length of 50 cm. The length, $x$ cm, of a rod is acceptable only if it satisfies $|x-50|\le0.8$; otherwise the rod is rejected.

   (a)

   (i) Solve the inequality $|x-50|\le0.8$, giving the acceptable range of rod lengths. [2]
   (ii) A second batch of rods, originally intended to be $x$ cm long, $x\ge0$, has actual lengths modelled by $M(x)=|2x-100|-3$. Find the values of $x$ for which $M(x)=5$. [4]

   [6 marks]

   (b)

   Let $p(x) = |x-50|$, for $x\ge50$.
   (i) Show that $p(x)=x-50$ for $x\ge50$, and explain why $p$ has an inverse function on this domain. [3]
   (ii) Find $p^{-1}(x)$, and determine whether the function $q(x)=|x-50|$, for $x\in\mathbb{R}$, is odd, even, or neither. Justify your answer. [3]

   [6 marks]

   Total for question 8: 12 marks

End of questions